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Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents
- 1. Background Model Baseline First Order Second Order Derivation Conclusion Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents Paolo Guasoni1,2 Marko Hans Weber3 Dublin City University1 Università di Bologna2 National University of Singapore3 Talks in Financial and Insurance Mathematics ETH Zürich, April 7th , 2022
- 2. Background Model Baseline First Order Second Order Derivation Conclusion Heterogeneity in Market Equilibrium • Market participants differ in preferences and endowments. • Can trade, borrow, and lend among themselves. • What is the implied dynamics of asset prices? • Does agents’ heterogeneity matter? • Which effects are more important? • Which agents survive? • Do agents trade?
- 3. Background Model Baseline First Order Second Order Derivation Conclusion Complete Markets • Endogenously complete markets. Anderson and Raimondo (2008), Hugonnier et al. (2012), Riedel and Herzberg (2013). • Equilibrium Pareto Optimal. • A representative agent exists. • Heterogeneity does not matter. • No dynamic trading. • Only one agent survives with power utility. • Endogenously incomplete markets? “with dynamic incompleteness, essentially nothing is known” (Anderson and Raimondo, 2008)
- 4. Background Model Baseline First Order Second Order Derivation Conclusion Market Incompleteness • Multiple reasons for incompleteness. • Unhedgeable shocks in individual labor incomes: Duffie et al. (1994), Christensen et al. (2012), Heaton and Lucas (1996). • Transaction costs: Vayanos (1998), Vayanos and Vila (1999), and Buss and Dumas (2019). • Asset restrictions for some agents: Basak and Cuoco (1998) and Guvenen (2009). • Stochastic fundamentals: Assets as necessary to hedge dividends’ shocks. Brennan and Xia (2001), Scheinkman and Xiong (2003), Dumas et al. (2009), Cujean and Hasler (2017).
- 5. Background Model Baseline First Order Second Order Derivation Conclusion Literature • Incomplete-market equilibria: Geanakoplos (1990), Marcet and Singleton (1999), Christensen, Larsen, Munk (2012, 2014). • Noise in fundamentals: Scheinkman and Xiong (2003), Dumas et al. (2009), Cujean and Hasler (2017). • No-trade theorems. Milgrom and Stokey (1982), Morris (1994), Feinberg (2000), Arieli et al. (2020). • Agents’ survival: Kogan et al., (2006). Cvitanic et al. (2012). • Markov equilibria: Duffie et al. (1994).
- 6. Background Model Baseline First Order Second Order Derivation Conclusion This Paper • Several long-lived agents: differ in risk-aversion and time-preference. • Many risky assets in unit supply, paying continuous dividend streams. Safe asset in zero net supply. • Assets’ fundamentals stochastic. Small, mean-reverting noise. • Zeroth-order equilibrium: Exact complete-market equilibrium without noise in fundamentals. • First-order equilibrium: small-noise expansion in closed form for prices, interest rate, consumption, trading policy. Representative agent. Predictability. Multifactor asset pricing. • Second-order equilibrium: No representative agent. Heterogeneity matters. Dynamic trading.
- 7. Background Model Baseline First Order Second Order Derivation Conclusion Preferences • n agents maximize lifetime utility from consumption. • The i-th agent has absolute risk-aversion αi and a time-preference rate βi . Maximizes E Z ∞ 0 e−βi s Ui (ci s)ds • Constant absolute risk aversion: Ui (c) = e−αi c −αi . • Agents finance consumption by: • Borrowing from and lending to each other • Earning dividends and trading the dividend-paying assets.
- 8. Background Model Baseline First Order Second Order Derivation Conclusion Market • One risky asset pays dividend stream Dt dDt = (µ̄ + εσzzt )dt + σDdWD t , dzt = −azt dt + dWµ t . • Wµ , WD independent Brownian motions. • zt fundamental variable. Parameter ε controls fluctuations’ size. • ε = 0: baseline complete market. • ε 0: incomplete market. Fluctuations unhedgeable: one stock, two shocks.
- 9. Background Model Baseline First Order Second Order Derivation Conclusion Budget Equation • i-th agent’s holds θi t shares of stock at time t. Consumes at rate ci t . • Budget equation: dXi t = θi t (dPt + Dt dt) + rt (Xt − θi t Pt )dt − ci t dt Xi 0 = xi • Wealth Xi t of i-th agent changes from • capital gains and losses θi t dPt ; • dividends θi t Dt dt; • interest income and charges rt (Xt − θi t Pt )dt; • consumption ci t dt. • Stock price Pt and interest rate rt endogenous.
- 10. Background Model Baseline First Order Second Order Derivation Conclusion Equilibrium 1 Exogenous Inputs • Agents’ preferences; • Distribution of wealth; • Dividends. 2 Equilibrium Conditions • Optimal consumption and investment; • Budget equation; • Market-clearing: supply one for stock, zero for safe asset, total consumption equals dividend. n X i=1 θi t = 1, n X i=1 (Xi t − θi t Pt ) = 0, n X i=1 ĉi t = Dt a.s. for all t 0 3 Endogenous Outputs • Prices; • Interest Rates; • Equilibrium consumption and Investment.
- 11. Background Model Baseline First Order Second Order Derivation Conclusion Baseline Model • When ε = 0, dividend growth rate is constant. • Aggregate risk-aversion ᾱ = 1/ Pn i=1 1 αi ; aggregate time-preference β̄ = Pn i=1 ᾱ αi βi . Theorem Assume β̄ + ᾱµD − ᾱ2 2 σ2 D 0. The pair (r, Pt ) defined by r = β̄ + ᾱµD − ᾱ2 2 σ2 D, Pt = Dt r + µD − ᾱσ2 D r2 is a market equilibrium, and the agents’ optimal policies are θ̂i t = ᾱ αi , ĉi t = β̄ − βi αi t − 1 r + ᾱ αi Dt + r xi − ᾱ αi P0 .
- 12. Background Model Baseline First Order Second Order Derivation Conclusion Baseline Features • Representative agent: same prices and rates as in equilibrium with one agent with preferences (ᾱ, β̄). • Markov Equilibrium: optimal policies and price dynamics depend only on (t, Dt ). • No-trade Equilibrium: number of shares constant for all agents. • Consumption affected by relative impatience and relative initial wealth.
- 13. Background Model Baseline First Order Second Order Derivation Conclusion Baseline Long Run Corollary In the long run (as t → ∞): 1 The proportion of total wealth Xi t Pt owned by the i-th agent converges to ᾱ αi + β̄−βi αi µD a.s. 2 The proportion of i-th agent wealth θi t Pt Xi t invested in stock converges to µD β̄−βi ᾱ +µD a.s. • All agents coexist indefinitely. Contrast to models with isoelastic investors, in which only one agent overtakes the others (most risk tolerant, patient, or combination thereof). Kogan et al. (2006), Yan (2008), Cvitanic et al. (2012), Guasoni and Wong (2020). • “Natural” wealth share and portfolio weight. Independent of initial wealth. • The more patient and risk tolerant, the wealthier.
- 14. Background Model Baseline First Order Second Order Derivation Conclusion First-Order Equilibrium – Definition Definition A first-order equilibrium is a family {rt (ε), Pt (ε), (ĉi t (ε))1≤i≤n, (θ̂i t (ε))1≤i≤n}0≤ε≤ε̄ such that (henceforth, unless there ambiguity arises, the dependence on ε is omitted) 1 Pt (ε)0≤ε≤ε̄ is a continuous semimartingale and rt (ε)0≤ε≤ε̄ an adapted, integrable process; 2 {ĉi t , θ̂i t }0≤ε≤ε̄ is admissible for all 1 ≤ i ≤ n; 3 For every ε the market clearing conditions hold: n X i=1 θ̂i t = 1, n X i=1 (Xi t − θ̂i t Pt ) = 0, n X i=1 ĉi t = Dt ; 4 For every agent i, the family of strategies {ĉi t , θ̂i t }0≤ε≤ε̄ is first-order optimal in ε, i.e., E Z ∞ 0 1 −αi e−βi t−αi ĉi t dt ≥ sup (ct ,θt )∈A E Z ∞ 0 1 −αi e−βi t−αi ct dt + o(ε).
- 15. Background Model Baseline First Order Second Order Derivation Conclusion First-Order Equilibrium – Result Theorem Assume r̄ 0. The following definition of interest rate rt , price Pt , consumption ĉi t and investment θ̂i t strategies yield a first-order equilibrium. For t ≤ Tε := −2 r̄ log ε, set P (1) t = Dt 1 r̄ − ε ᾱ r̄(a + r̄) σzzt + (µ̄ − ᾱσ2 D) 1 r̄2 − ε 1 r̄ + 1 a + r̄ ᾱσz r̄(a + r̄) zt + ε 1 r̄(a + r̄) σzzt , r (1) t = r̄ + εᾱσzzt , θ̂i t = ᾱ αi , ĉi t = ᾱ αi Dt + β̄ − βi αi t − 1 r̄ + ε ᾱσz (a + r̄)2 z0 + r̄ + ε ᾱr̄ a + r̄ σzz0 xi − ᾱ αi P0 − ε αi Z t 0 mi,1 s dWµ s + ε2 mi,1 t = −αi ᾱσz a + r̄ β̄ − βi αi t + 1 a + r̄ + xi − ᾱ αi P0 r̄ while for t Tε the first-order equilibrium is as in the baseline.
- 16. Background Model Baseline First Order Second Order Derivation Conclusion Price and Rate Dynamics • Procyclical interest rate. Follows Vasicek model. r (1) t = r̄ + εᾱσzzt , • Good times (zt high) have twofold effect: • decrease prices by discount effect; • Increase prices by dividend growth. P (1) t = Dt 1 r̄ − ε ᾱ r̄(a + r̄) σzzt + (µ̄ − ᾱσ2 D) 1 r̄2 − ε 1 r̄ + 1 a + r̄ ᾱσz r̄(a + r̄) zt + ε 1 r̄(a + r̄) σzzt , • Preferences enter only through ᾱ and β̄. Representative agent persists at first order. • State variables Dt and zt . Markov equilibrium?
- 17. Background Model Baseline First Order Second Order Derivation Conclusion Agents’ Choices • Stock holdings: θ̂1,i t = ᾱ αi . No-trade equilibrium at first order. • Consumption: ĉ1,i t = ᾱ αi Dt + β̄ − βi αi t − 1 r̄ + ε ᾱσz (a + r̄)2 z0 + r̄ + ε ᾱr̄ a + r̄ σzz0 xi − ᾱ αi P0 + ε Z t 0 β̄ − βi αi s + 1 a + r̄ + xi − ᾱ αi P0 r̄ ᾱ a + r̄ σzdWµ s + ε2 Rt • Agents need (t, Dt , zt , Wµ t , R t 0 sdWµ s ) to consume. Markov equilibrium on these variables. • Few public signals sufficient for all agents. • Positive shock to dividend growth: more consumption for natural lenders (xi ᾱ αi P0) and patient agents (βi β̄). • Agents compensate through decentralized consumption coordination.
- 18. Background Model Baseline First Order Second Order Derivation Conclusion Bonds’ Shadow Prices • Long-term bonds do not exist in this market. • Consol bond: pays a constant unit coupon (dt). Linear bond: pays a linear coupon tdt. • Shadow price: hypothetical price for which some agent’s demand would be zero. A priori, depends on agent. • A posteriori: Ci,1 t = 1 r̄ 1 − ε ᾱ a + r̄ σzzt + o(ε), (consol bond) Li,1 t = t · Ct + 1 r̄2 − ε 1 r̄ + 1 a + r̄ ᾱσz r̄(a + r̄) zt + o(ε) (linear bond) • Agents agree on shadow bond prices. Bonds are first-order “useless”. • Bond prices explain effect of shocks on stock prices.
- 19. Background Model Baseline First Order Second Order Derivation Conclusion Return Predictability • Define excess expected cash flow as Es,t P := E Pt e− R t s rudu + Z t s Due− R u s rl dl du
- 23. Fs # − Ps. • Returns are unpredictable if Es,t P is deterministic (new information does not help forecasting). • Instead, the model implies Es,t P = ᾱσ2 D r̄2 (1−e−r̄(t−s) )+εzs ᾱ2 σ2 Dσz ar̄2(a + r̄)2 e−r̄(t−s) (a + r̄)2 − e−(a+r̄)(t−s) r̄2 − a2 − 2ar̄ +o(ε) • Fundamentals zs help forecasting. But one should not trade! • In good times expected excess cash flows are lower. Discounting effect prevails.
- 24. Background Model Baseline First Order Second Order Derivation Conclusion Multiple Assets • Model extends to several stocks: dDt = (µ̄ µ µ + εσz σz σz z z zt )dt + σD σD σD dWD t , dzt = −azt dt + ρ ρ ρ dWD t + q 1 + |ρ ρ ρ |2dW̃µ t . where WD and Wµ are k-dimensional independent Brownian Motions, and σz σz σz, A ∈ Rk×k . • Cross-sectional asset pricing implication: At the first order, Ri =βi Rm +ε ᾱ(1T σD σD σD ρ ρ ρ) r̄2(a + r̄) (1T σz σz σz) −ᾱ Di t − βi (1T Dt ) + σz σz σz i 1T σz σz σz − βi − 2a + 3r̄ r̄(a + r̄) ᾱ µ µ µi −βi (1T µ µ µ) . • βi is the market beta of the i-th asset. CAPM holds if ρ ρ ρ = 0. Deviations otherwise. • Fixed value effect from relative growth µ µ µi −βi (1T µ µ µ). • Relative dividend Di t − βi (1T Dt ) produces size effect. No low-dimensional factor models.
- 25. Background Model Baseline First Order Second Order Derivation Conclusion Second-Order Equilibrium • Similar construction as for first order. Optimality must hold up to o(ε2 ) errors. • New features. • Trading arises. θ̂i not constant. • Equilibrium does not reduce to representative agent. Heterogeneity matters. • Interest-rate risk premium. Risk-neutral vs. physical dynamics. • Changes to long-term means.
- 26. Background Model Baseline First Order Second Order Derivation Conclusion Interest Rate r (2) t = r (1) t − ε2 2 n X j=1 ᾱ αj (mj,1 t )2 • Autarchic endowments (xi = ᾱ αi P0) lead to a rate-decrease proportional to n X i=1 ᾱ αi (β̄ − βi )2 • Cross-sectional variance of time-preference, weighted inversely to risk aversion. • With homogeneous time-preference, rate-decrease proportional to n X i=1 αi xi − ᾱ αi P0 2 • Cross-sectional variance of endowment deviations, weighted by risk aversion.
- 27. Background Model Baseline First Order Second Order Derivation Conclusion Asset Prices P (2) t = P (1) t + ε2 Dt C̃ (2) t + (µ̄ − ᾱσ2 D)L̃ (2) t − 2ᾱσ2 z r̄(a + r̄)(2a + r̄) z2 t + 1 r̄ ! , • C̃ (2) t , L̃ (2) t : cross-sectional averages of second-order corrections to bonds’ shadow prices: C̃ (2) t = N X j=1 ᾱ αj Cj,2 t L̃ (2) t = N X j=1 ᾱ αj (Lj,2 t − t · Cj,2 t ) Ci t = Ci,1 t + ε2 Ci,2 t + o(ε2 ), Li t = Li,1 t + ε2 Li,2 t + o(ε2 ). • Bond prices corrections explain first two terms through discounting. • Last term reflects effect of the state, net of discounting.
- 28. Background Model Baseline First Order Second Order Derivation Conclusion Dynamic Trading θ̂i t = ᾱ αi + ε2 r̄ σ2 D mi,1 t αi pm1 t = ᾱ αi + ε2 ᾱ2 σ2 z (a + r̄)2σ2 D β̄−βi αi t + 1 a+r̄ + xi − ᾱ αi P0 r̄ Dt + (a+2r̄)(µ̄−ᾱσ2 D) r̄(a+r̄) − 1 ᾱ • Why do agents trade? • Because the young buy what the old sell? No, agents live forever. Overlapping generations models: Vayanos (1998), Vayanos and Vila (1999). • To hedge fluctuations in personal labor incomes? No, labor incomes are absent. Heaton and Lucas (1996), Lo et al. (2004), Christensen et al. (2012), Herdegen and Muhle-Karbe (2018), and Buss and Dumas (2019). • To hedge fundamental shocks in an incomplete market. Hedging strategies cannot account for all future contingencies and need updating. • Note that trading strategies depend only on mi,1 t and sum to zero. No price pressure.
- 29. Background Model Baseline First Order Second Order Derivation Conclusion Consumption • Net of dynamic trading adjustments, the second-order effect on consumption is: ε2 2αi (mi,1 t )2 − N X j=1 ᾱ αj (mj,1 t )2 • Everyone wants a reward for additional consumption volatility (mi,1 t ). • But not everyone can get it: sum must be zero. • Instead, everyone is rewarded for above-average consumption volatility. • Atypical agents (large mi,1 t ) get paid. Typical agents (small mi,1 t ) pay.
- 30. Background Model Baseline First Order Second Order Derivation Conclusion Agreement on Traded Assets • Marginal utility of i-th agent: e−βi t−αi ĉi t = yi Mi t • Stochastic discount factor for ε = 0 is Mt = e−β̄t−ᾱ(Dt −D0) . • Set Mi t = e−β̄t−ᾱ(Dt −D0)+Yi t , where limε↓0 Yi t = 0 1 αi −β̄t − ᾱ(Dt − D0) + Yi t = − βi αi t − ĉi t − 1 αi log yi , • If Yi t = εYi,1 t + o(ε), where dYi,1 t = li,1 t dt + mi,1 t dWµ t + ni,1 t dWD t . Then li,1 t = ni,1 t = 0. • Follows from the condition that all agents agree on the interest rate and the stock price. • How to identify mi,1 t ?
- 31. Background Model Baseline First Order Second Order Derivation Conclusion Replicability of Consumption Stream • Consumption stream from first-order condition: ĉi t = ᾱ αi (Dt − D0) + β̄ − βi αi t − 1 αi log yi − 1 αi Yi t • Price both sides and set Li t = Et hR ∞ t s Mi s Mi t ds i , Ci t = Et hR ∞ t Mi s Mi t ds i , and Ki t = Et hR ∞ t Yi s Mi s Mi t ds i . Xi t = ᾱ αi Pt + β̄ − βi αi Li t − ᾱ αi D0Ci t − 1 αi (log yi )Ci t − 1 αi Ki t . • Must satisfy budget equation dXi t = θi t (dPt + Dt dt) + rt (Xi t − θi t Pt )dt − ci t dt • Matching diffusion coefficients of Xi t from both equations yields expressions for mi,1 t and θi,1 t .
- 32. Background Model Baseline First Order Second Order Derivation Conclusion Conclusion • Incomplete-market equilibrium with heterogeneous preferences. • Incompleteness from unhedgeable fundamental shocks. • Perturbation around complete baseline model. • First- and second-order equilibria. • Uncover first- and second-order effects. • First order: representative agent, no trading, four-state Markov, multifactor pricing. • Second Order: Dynamic trading, heteregeneity, premium for above-average consumption risk.
- 33. Background Model Baseline First Order Second Order Derivation Conclusion Duality Lemma For every admissible consumption process ct and initial wealth x, 1 −αi E Z ∞ 0 e−βi s Ui (cs)ds ≤ − Ci 0 αi exp ( −E R ∞ 0 Mi t log Mi t dt − βi Li 0 − xαi + o(ε) Ci 0 ) . • Duality lemma: upper bound for all strategies. • Explicit formulas for Mi t and ĉi t : upper bound attained. • Optimality at first order follows. • Second-order result analogous.
- 34. Background Model Baseline First Order Second Order Derivation Conclusion Thank You! Questions? http://ssrn.com/abstract=4057156
- 35. Background Model Baseline First Order Second Order Derivation Conclusion Admissible Strategy Definition Let Pt (ε) be a continuous semimartingale and rt (ε) an adapted, integrable process, for all ε ∈ (0, ε̄). A family {ct (ε), θt (ε)}0≤ε≤ε̄ 0≤t∞ is an admissible strategy if 1 (ct (ε), θt (ε)) is a pair of adapted processes for all ε ∈ (0, ε̄). 2 There exists δ 0 independent of ε such that for any ε ∈ (0, ε̄) and any T the process θt (ε) is (2 + δ)-integrable on [0, T] × Ω. 3 The process ct (ε) is bounded from below uniformly in ε, i.e, for some polynomial functions q1, q2 independent of ε, ct (ε) ≥ q1(t, Dt ) + R t 0 q2(s, Ds)dWD s . 4 For every ε ∈ (0, ε̄) the budget equation dXt (ε) = θt (ε)(dP(ε) + Dt dt) + rt (ε)(Xt (ε) − θt (ε)Pt (ε))dt − ct (ε)dt, X0(ε) = xi , has a strong solution Xt (ε) that is bounded from below uniformly in ε, i.e., for some polynomials p1, p2, p3 independent of ε, Xt (ε) ≥ p1(t, zt , Dt ) + R t 0 p2(s, zs, Ds)dWD s + R t 0 p3(s, zs, Ds)dWµ s , and is (2 + δ)-integrable on [0, T] × Ω for some δ 0 independent of ε and any T.